Reconceptualizing Mathematics: Courses for Prospective and Practicing Teachers
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Reconceptualizing Mathematics: Courses for Prospective and Practicing TeachersSusan D. NickersonMichael Maxon San Diego State University


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Teachers matter. Practicing TeachersFrom a carefully chosen sample of 2525 adults, representing a cross-section of U. S. adults, an overwhelming majority agreed that improving the quality of teaching was the most important way to improve public education.


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Every Child Mathematically Proficient Practicing Teachersfrom theLearning First Alliance, 1998

Before It’s Too Late: A Report to the Nation from theNational Commission on Mathematics and Science Teaching for the 21st Century, 2000

What Matters Most: Teaching for America’s Future and No Dream Denied from the National Commission on Teaching and American’s Future, 1996 & 2003

The Mathematical Education of Teachers from the Conference Board of the Mathematical Sciences, 2001

Mathematical Proficiency for All Students from the RAND Mathematics Study Panel, 2003

Educating Teachers of Science, Mathematics, and Technology from the National Research Council, 2001

Undergraduate Programs and Courses in the Mathematical Sciences from MAA


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Consider the following: Practicing Teachers A teacher wants to pose a question that will show him whether his students understand how to put a series of decimals in order from smallest to largest. Which of the following sets of decimal numbers will help him assess whether his pupils understand how to order decimals? Choose each that you think will be useful for his purpose and explain why. .123 1.5 2 .56 .60 2.53 3.14 .45 .6 4.25 .565 2.5Or would each of these work equally well for this purpose?


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There is a growing understanding that the mathematics of the elementary and middle school is not trivial, and that teachers need more preparation and different preparation than has been common.


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Looking ahead elementary and middle school is not trivial, and that teachers need more preparation and different preparation than has been common.RecommendationsOur focus with examples of content and instructional possibilitiesGuiding QuestionsExamples of our PD structures and delivery


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In 1998, the National Research Council (NRC) appointed a committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.


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The Committee’s 2001 report, committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.Educating Teachers of Mathematics, Science, and Technology, includes recommendations about the characteristics teacher education programs in mathematics and science should exhibit.


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Programs should have the following features: committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.• Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers;

• Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach;

• Teach content through the perspectives and methods of inquiry and problem-solving


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Programs should have the following features: committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.• Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers;• Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach;• Teach content through the perspectives and methods of inquiry and problem-solving


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Collaborative endeavors are a part of our: committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.• Pre-service courses

• Professional development:

summer courses, usually at the university (3 days to 2 weeks)

partnership with several districts, urban & rural

classes through the university

on-line hybrid courses


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Integrating Learning Mathematics with Practice committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.

Director of Mathematics

Site Mathematics Administrators

Mathematics Instructors

(SDSU)

Math Teacher Ed Instructors (SDSU)

Teachers inprogram

All Teachers at site

Math Resource Teacher


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Programs should have the following features: committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.• Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers;• Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach;• Teach content through the perspectives and methods of inquiry and problem-solving


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What content preparation do teachers need to have to teach middle school mathematics well? The 2001 document from CBMS, The Mathematical Education of Teachers recommends that for teaching middle school mathematics: At least 21 semester hours of mathematics, some of which should include a study of elementary mathematics, some of which should focus on the middle grades, and some of which include the study of mathematics proper.


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Content recommendations in MET are made in four areas: middle school mathematics well? 1. Number and Operations2. Algebra and Functions3. Measurement and Geometry4. Probability and Statistics


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Number and Operations middle school mathematics well? • Understand and be able to explain the mathematics that underlies the procedures used for operating on whole numbers and rational numbers.


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Example 1: Multiplication of Fractions middle school mathematics well? “Juanita had mowed 4/5 of the lawn, and her brother Jaime had raked 2/3 of the mowed part. What part of the lawn had been mowed and raked?”


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Example 2: Why do we invert and multiply when dividing fractions?

Write a story problem that can be represented and solved by

2 1/2 ÷ 1/3

  • First, divide by the unit fraction: 1 ÷ 1/3

    • 1 ÷ 1/3 is 3, or 3/1

    • So 2 ÷ 1/3 is twice 3, or 6, or

    • in general k ÷ 1/n = k n


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2. Algebra and Functions fractions?• Understand and be able to work with algebra as a symbolic language, as a problem solving tool, as generalized arithmetic, as generalized quantitative reasoning, as a study of functions, relations, and variation, and as a way of modeling physical situations.


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2. Algebra and Functions fractions?• Understand and be able to work with algebra as a symbolic language, as a problem solving tool, as generalized arithmetic, as generalized quantitative reasoning, as a study of functions, relations, and variation, and as a way of modeling physical situations.


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Coping strategies: fractions?1. Just add2. Guess at the operation to be used


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Limited Strategies fractions?3. Look at the number sizes and use those to tell you which operation to use.4. Try all operations and choose the most reasonable answer.5. Look for “key words.”6. Decide whether the answer should be larger or smaller than the given numbers, then decide on the operation.


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Desired strategy: fractions?7. Choose the operations with the meaning that fits the story. Perhaps draw a picture to help understand the problem.


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Consider this problem: fractions?Dieter A: I lost 1/8 of my weight. I lost 19 pounds.Dieter B: I lost 1/6 of my weight, and now you weigh 2 pounds more than I do.How much weight did Dieter B lose?


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First, make a list of relevant quantities fractions?

  • Dieter A’s weight before the diet

  • Dieter A’s weight after the diet

  • Fraction of weight lost by A

  • Amount of weight lost by A

  • Difference in weight of A and B before diet

  • Difference in weight after diet

  • Etc.


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Now we consider the values of the quantities fractions?

  • Dieter A’s weight before the diet: ?

  • Dieter A’s weight after the diet: ?

  • Fraction of weight lost by A: 1/8

  • Amount of weight lost by A: 19 pounds

  • Difference in weight of A and B before diets: ?

  • Difference in weights after diets: 2 pounds (A is 2 pounds less than B

  • Etc.


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Dieter A’s original weight (before diet) fractions?

Before diet

Shows weight loss of 1/8 of original diet. Weight lost was 19 lb.

After diet


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A’s weight before diet: 19 x 8 = 152 fractions?A’s weight after the diet 152–19=133B’s weight after the diet: 133 + 2 = 135135 is 5/6 of B’s weight before the diet so B weighed 162 poundsB lost 162 – 135 = 27 pounds


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2. Algebra and Functions fractions?• Recognize change patterns associated with linear, quadratic, and exponential functions.Example: Growing Dots


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3. Measurement and Geometry fractions?• Identify common two-and three dimensional shapes and list their basic characteristics and propertiesExample here GSP quadrilateralsvenn


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4. Data Analysis, Statistics, and Probability fractions?• Draw conclusions with measurements of uncertainty by applying basic concepts of probabilityEx: three card poker


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Programs should have the following features: fractions?• Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers;• Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach;• Teach content through the perspectives and methods of inquiry and problem-solving


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Cynthia has 1 cup of sugar and each recipe requires: a) 1/2, b) 1/3, c)2/3, d) 3/4, e) 4/5 of a cup of sugar. How many recipes can she make?Model and draw each case. What number sentence describes each?Later generalize for more or less than one cup of sugar.


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In a report by the MAA Committee on the Undergraduate Preparation in Mathematics (2004):“..does not mean the same thing as preparation for the further study of college mathematics. For example, while prospective teachers need knowledge of algebra..the traditional college algebra course with primary emphasis in developing algebra skills does not meet the needs of elementary [and middle school] teachers.”


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Looking ahead Preparation in Mathematics (2004):RecommendationsOur focus with examples of content and instructional possibilitiesGuiding QuestionsExamples of our PD structures and delivery


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Decisions about what to include rely on answers to guiding questions

• Pre-service courses• Professional development:

summer courses, usually at the university (3 days to 2 weeks)

partnership with districts

classes through the university

on-line hybrid courses


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Guiding Questions: questions

What is the content at their grade level? With what have they had only superficial exposure? What would be difficult to learn from the curriculum?

How does this content need to be extended?

How does the research literature characterize the difficulties & student misconceptions?

Does this group of teachers have particular needs?


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What is the content at their grade level? With what have they had only superficial exposure? What would be difficult to learn from the curriculum?

Ratio and proportional reasoning

Minimal and sufficient definitions of quadrilaterals


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Procedures can sometimes be learned from the curriculum. When we talk about content we are talking about conceptual understanding and procedural fluency.


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2) How does this content need to be extended? When we talk about content we are talking about


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3) How does the research literature characterize the difficulties & K-12 student misconceptions?

Specialized knowledge for teaching


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Difficulties in algebra include difficulties & K-12 student misconceptions?

  • Misunderstanding the equals sign

  • Comprehending use of literal symbols as generalized numbers or variables

  • Expressing relationships in a variety of ways such as tables, graphs, and equations

  • Understanding the role of the unit

  • Ratio and Proportional reasoning


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Misunderstanding the equals sign difficulties & K-12 student misconceptions?

Students tend to misunderstand the equal sign as a signal for “doing something” rather than a relational symbol of equivalence.


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Comprehending the many uses of literal symbols difficulties & K-12 student misconceptions?

  • A=L x W

  • 40=5x

  • 2a + 2b = 2(a + b)

  • Y = 3x + 5


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Expressing relationships in a tables, graphs, and equations difficulties & K-12 student misconceptions?


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Understanding the unit difficulties & K-12 student misconceptions?


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Ratio and Proportional Reasoning difficulties & K-12 student misconceptions?

Proportional reasoning has been called a watershed concept, the “capstone of elementary arithmetic and the cornerstone of all that is to follow”

Situations that call for understanding the close relationship between fractions and ratios, both of which are represented by the same notation, cause particular problems.


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A donut machine produces 60 donuts every 5 minutes. How many donuts does it produce in an hour?a. Identify the rate in this problem. What unit rate is associated with this rate? b. Write a proportional statement that could be used to find the number of donuts produced in an hour.c. Made a graph of the donuts produced every 5 minutes and a graph of the donuts produced every minute. Are the graphs different? Why or why not?d. What is the slope of each line?


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Spanning the curriculum donuts does it produce in an hour?

Students must have an understanding of how to write a useable mathematical definition and establish equivalence among definitions.

Students develop a disposition for problem solving.

Students must learn to use valid reasoning to reach and justify conclusions.



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Looking ahead donuts does it produce in an hour?RecommendationsOur focus with examples of content and instructional possibilitiesGuiding QuestionsExamples of our PD structures and delivery


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Pre-service courses for middle school math teachers: donuts does it produce in an hour?

Min 32 units; 3 in each content area

5 courses for elementary:

Number & Op (3)

Algebra (2 or 3)

Geo & Meas (3)

Prob & Stats (2 or 3)

Children’s mathematical thinking (1.5)

2 courses for middle school:

Math for Middle School Teach (3)

Transition to Higher Math or History of Math (3)

Other courses:

Calculus I

Transition to Higher Math or History of Math (3)

Statistics (3)

Number theory (3)


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Math for Middle School Teachers donuts does it produce in an hour?

Pre-service course

Guided by NCTM Curriculum Focal Points (Gr 6-8)

• Division of Fractions

• Operations with Integers

• Ratio & Proportion

• Functions and relations

Children’s thinkingin these same content areas


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Professional development courses donuts does it produce in an hour?

• Immediacy. Teachers back in classes tomorrow, bring questions about their own experiences and students.

• Collaborative team encompasses school administrators and educators with a focus on scaffolding of ideas

• Usually more flexibility to design course particular to group/team of teachers. Connections tight among concepts.


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Designing courses for professional development donuts does it produce in an hour?

• Summer courses

• Partner with school district 1 to offer 4 courses to qualify teachers for NCLB

• Partner with School District 2 to provide PD to 7th through Alg 1 in pull out days (only 30 hours) each of 2 years


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Designing courses for professional development donuts does it produce in an hour?

• Partner with School District 3 to provide PD to middle school (either pull-out or after school)

• Partner with School District 4 to provide PD to 7-12 (summer and after school)

• On-line planned for middle school


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• Teach content so they know well, understand deeply, and use effectively the fundamental content they teach

• Teach content through the perspectives and methods of inquiry and problem-solving

It’s not just what we teach teachers,

but how we teach teachers.


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Discussion use effectively the fundamental content they teach


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That students do not regard algebra as a sense-making and useful subject is often due to the way that algebra is often taught and the way that students are prepared for algebra.Middle school teachers themselves must make sense of algebra and its underpinnings.


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“Algebra has been experienced as an unpleasant, even alienating event, mostly about manipulating symbols that do not stand for anything. (But) algebraic reasoning in its many forms, and the use of algebraic representations such as graphs, tables, spreadsheets and traditional formulas, are among the most powerful intellectual tools that our civilization has developed.” (Kaput 1999)


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